As Mike Shulman reminded us, spectra have Postnikov towers; they also have Moore towers because they’re naturally pointed things. We here on this blog used Moore towers to build a Leray+Serre-type spectral sequence for cohomology; just for variety, let’s see what Postnikov says about homology.

For an $\Omega^\infty$-space $A$ and a fibration

\[ F \to E \to X \] \[ E \simeq \coprod_{x:X} F_x \] there is a derived dependent $\Omega^\infty$-space

\[ \left( x:X \mapsto \th(F;A) \right) : X \to \Omega^\infty .\]

The first thing that wants doing is to compare the homology of $X$ in this dependent $\Omega^\infty$-space with homology of $\coprod_{x:X} F_x$ in $A$; but this is straight-forward: the map $ \Th^{(k)}(F;A) = \Th(F;A^{(k)}) \to \th^{(k)}(F;A) $ is $2k$-ish-connected, and so the map

\[ \Th (X; \Th^{(k)}(F;A) ) \to \Th(X; \th^{(k)}(F;A) ) \] is also $2k$-ish-connected; but

\[ \Th( X ; \Th^{(k)}(F;A) ) = \Th( \textstyle\coprod\limits_{x :X } F_x ; A^{(k)}) = \Th^{(k)}(E;A ) \] so that

\[ \colim_k \Omega^k \Th^{(k)}(E;A) \simeq \colim_k \Omega^k \Th^{(k)}(X ; \th (F; A)) \simeq \colim_k \Omega^k \Th(X ; \Th^{(k)}(F,A)) \] all describe the same $\Omega^\infty$-space.

With that out of the way, we can now use the coefficient long exact sequence together with whatever filtration of things makes for easy calculations. I.i.r.c., with $\Omega^\infty$ maps of $\Omega^\infty$ spaces, all fibrations are principal, so Postnikov towers actually correspond to Postnikov *systems*; that is, we have a chain of fiber sequences

\[ \| \th(F,A) \|_0 = \pi_0 ( \th(F,A) ) = K( H_0 (F,A), 0) \] \[ \|\th (F,A)\|_{r+1} \to \|\th(F,A)\|_{r} \to K( H_{r+1}(F,A), r+2) \] and convergence

\[ \lim_r \| \th (F,A) \|_{r} = \th(F,A) .\] The long fiber sequences for $\th(F,A)$ aren’t themselves terribly interesting, but they induce (by the coefficient long exact homology sequence, from last time) fiber sequences of twisted homology spectra:

\[ \th (X ; \|\th(F,A)\|_{r+1}) \to \th(X;\|\th(F,A)\|_r) \to \th(X;K(H_{r+1}(F,A),r+2) ) \]

and therefore long exact sequences of groups

\[ H_q ( X ; \|\th(F;)\|_{r+1} ) \to H_q(X;\|\th(F;)\|_{r}) \to H_{q-r-2}(X, \mathcal{H}_{r+1} ( F; ) ) \to H_{q-1} (X; \|\th(F; )\|_{r+1} ) \] extracting the first differential, this gives a spectral sequence with a page

\[ E^{(2)}_{p,s} = H_p ( X ; \mathcal{H}_s (F ; A ) ) \] and differential

\[ H_{p+2} ( X ; \mathcal{H}_{s} (F, A) ) \overset{d_2}\to H_p (X;\mathcal{H}_{s+1} (F,A) \]

It might (or might not) be a fun game to see why the Moore-Postnikov tower gives the same $E^{(2)}$ page with the same *type* differential, though I don’t know if it can/must/does be *the same* differential…

But what I actually think ought to be done *next* is relations between cohomology and homology. I think we want a spectral coefficient theorem.